The Riesz maps provide building blocks for developing preconditioners for more
complex systems [1]}, [2]}, [3]}, [4]}. In this final
example, we demonstrate this by constructing preconditioners for the mixed
formulation of the Hodge Laplacians associated with the \(L^2\)  de Rham
complex. For \(\Gamma _D = \emptyset \) , the problem is to find \((\sigma , u) \in V^{k-1}\times V^k\)  such
that
\(-(\tau , \sigma )_\Omega + (\operatorname{d}^{k-1}\tau , u)_\Omega &= 0 &&\quad \forall \,\tau \in V^{k-1},\\(v, \operatorname{d}^{k-1}\sigma )_\Omega + (\operatorname{d}^k v, \operatorname{d}^k u)_\Omega &= F(v) &&\quad \forall \,v \in V^k,\) 
